Ph.D. Thesis - Physics

Figure 7-3: Calculated secular frequencies for the trap at 3.5 MHz and a series of rf amplitudes.
The frequencies are blue, cyan, and green for x, y, and z respectively, with
ωˆz > ωˆx > ωˆy
PCB traps, it is on the order of 100 µm.
7.1.2 Ion crystal structure
The structure of ion crystals is computed by numerically minimizing the potential energy
of a given number N of trapped ions. We intuitively expect the crystals to align in the
ˆx-ˆy plane. As the number of ions increases, the potential energy can be lowered by the ion
crystal extending itself along ˆz. However, we expect that since the vertical confining fields
are still roughly twice as strong as the horizontal fields, the extent along ˆx and ˆy should
still be larger than that along ˆz, so that perhaps the ion crystal will still be approximately
planar.
We calculate the ion crystal structure using the Mathematica notebook included in
Appendix B. We first check this program against the analytical value for the separation of
only two ions, which is simple to compute by balancing the Coulomb and trap forces (see
Eq. 7.2). The ions are expected to align along the direction of weakest secular frequency,
which is ˆy. The analytical value for ωˆy/(2π) = 151 kHz is 16.12 µm. From the Mathematica
code, the value is 16.16 µm. This gives an idea of the inaccuracy level of the numerical
calculation; it is quite close to the analytical value.
In the calculations below, we assume the motional frequencies calculated above for Vrf
= 150 V. We first study the structure of 2-D crystals, and then observe (in theory at least)
the expansion of the crystal into the vertical direction as the number of ions is increased.
Fig. 7-5 shows the structure of 2-D and 3-D ion crystals for four, seven, and 15 ions. Fig. 7-6
is the same for 30 and 60 ions.
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